Sufficient Conditions for Hamiltonicity of Graphs with Respect to Wiener Index, Hyper-Wiener Index, and Harary Index

Abstract

In this paper, with respect to the Wiener index, hyper-Wiener index, and Harary index, it gives some sufficient conditions for some graphs to be traceable, Hamiltonian, Hamilton-connected, or traceable for every vertex. Firstly, we discuss balanced bipartite graphs with , where is the minimum degree of G, and gain some sufficient conditions for the graphs to be traceable or Hamiltonian, respectively. Secondly, we discuss nearly balanced bipartite graphs with and present some sufficient conditions for the graphs to be traceable. Thirdly, we discuss graphs with and obtain some conditions for the graphs to be traceable or Hamiltonian, respectively. Finally, we discuss t-connected graphs and provide some conditions for the graphs to be Hamilton-connected or traceable for every vertex, respectively.

1. Introduction

We consider only a simple graph. For a graph , we use n to denote and to . We write as the minimum length of the paths between x and y in G, which is called the distance between two vertices x and y of G. Denote . For a bipartite graph , if , it is called a balanced bipartite graph; if , it is called a nearly balanced bipartite graph. For disjoint graphs and , the graph is called the union of and , where and ; the graph is called the join of and , where and . For example, . The graph means the complement of G, where and . The graph , is called the quasicomplement of bipartite .

In [1], Wiener introduced the Wiener index, , for a connected graph G in 1947.

We denote ; then,

For a connected graph G, is called its hyper-Wiener index, which is introduced by Klein et al. [2] in 1995.

We denote then,

In [3, 4], Plavšić et al. and Ivanciuc et al. gave the Harary index, for a connected graph G, independently.

We denote ; then,

For a graph G, if it contains a path (cycle) containing all vertices of G, it is traceable (Hamiltonian); if there are paths containing all vertices of G between any two vertices in G, it is called to be Hamilton-connected; if there are paths containing all vertices of G from any vertex in G, it is called to be traceable from every vertex.

Topological indices have attracted much attention in the literature, and they are widely applied in various fields of science and technology. Recently, some interesting results have been obtained, see [5–11]. Especially, some topological indices are used to characterize the Hamiltonian property of graphs. We refer readers to see [2, 12–21]. Among them, Hua and Ning [15] gave Hamiltonian conditions for a balanced bipartite graph with respect to the Wiener index and Harary index. Cai et al. [13], by the hyper-Wiener index, gave conditions for a balanced bipartite graph to be traceable and Hamiltonian and a t-connected graph to be Hamiltonian, respectively. Li [18, 19] gave Hamitonian conditions for a t-connected graph with respect to the Wiener index and Harary index, respectively. With the Wiener index, hyper-Wiener index, and Harary index, Yu et al. [20] gave conditions for a t-connected graph to be Hamilton-connected and then got conditions for it to be traceable from every vertex, as well as gave conditions for a nearly balance bipartite graph to be traceable. Especially, Liu et al. [16, 17] studied the Hamiltonian property of graphs with respect to the Wiener index and Harary index of complement of graph or quasicomplement of bipartite graph, respectively. As a continuance of these results, we also study the similar problems.

In this paper, we discuss the hamiltonicity of graph by the Wiener index, hyper-Wiener index, and Harary index for quasicomplement or complement. In Section 2, we present some notations and some lemmas needed in the following. In Sections 3 and 4, we present some conditions for a balanced bipartite graph G with to be traceable and Hamiltion, respectively. In Section 5, we give sufficient conditions for a nearly balanced bipartite graph G with to be traceable. In Sections 6 and 7, we present some conditions for a graph G with to be traceable and Hamiltonian, respectively. In Sections 8 and 9, we provide some conditions for a t-connected graph to be Hamilton-connected and traceable from every vertex, respectively.

2. Preliminaries

Especially, in the following, for bipartite graphs, we always fix their partite sets. For instance, and are seen as different bipartite graphs unless .

A graph is the graph obtained from by deleting all possible edges between and and possible edges between and , where be bipartite graphs. Next, we denote some special classes of graphs:

Note that , and is not traceable.

Lemma 1. Denote as a quasicomplement of bipartite G with vertices. If is a connected balanced bipartite graph, then

Proof. As is a quasicomplement,

Lemma 2. Let be a quasicomplement of bipartite G with vertices. If is a connected balanced bipartite graph, then

Proof. As is a quasicomplement,

Lemma 3. Let be a quasicomplement of bipartite G of order . If is a connected balanced bipartite graph, then

Proof. As is a quasicomplement,

Lemma 4. Let be a complement of G with n vertices. If is a connected graph, then

Proof.

Lemma 5. Let be a complement of G with n vertices. If is a connected graph, then

Proof.

Lemma 6. Let be a complement of G with n vertices. If is a connected graph, then

Proof.

3. Traceable of Balanced Bipartite Graphs

Lemma 7 (see [22]). Let G be a balanced bipartite graph with vertices. If , , andfor some integer t, then G is traceable or or , .

Theorem 1. Let be a connected balanced bipartite graph with vertices. If , , andfor some integer t, then G is traceable or and .

Proof. Since , by Lemma 1, we get . By Lemma 7, we obtain that G is traceable or or , .
If : because is connected and , we get It is in contradiction with the above.
If : because is connected, .

Theorem 2. Let be a connected balanced bipartite graph with vertices. If , , andfor some integer t, then G is traceable or , .

Proof. Since , by Lemma 2, By Lemma 7, G is traceable or or , . By the same discussion as the proof of Theorem 1, the conclusion is established.

Theorem 3. Let G be a connected balanced bipartite graph with vertices. If , , andfor some integer t, then G is traceable or , .

Proof. Since , by Lemma 3, we get By Lemma 7, we obtain that G is traceable or or , . By the same discussion as the proof of Theorem 1, the conclusion is established.

4. Hamiltonian of Balanced Bipartite Graphs

Lemma 8 (see [23]). Let G be a balanced bipartite graph with vertices. If , andfor some integer t, then G is Hamiltonian or .

Theorem 4. Let be a connected balanced bipartite graph with vertices. If , , andfor some integer t, then G is Hamiltonian.

Proof. Since , by Lemma 1, we get . By Lemma 8, G is Hamiltonian or .
If : because is connected and , we get . It is in contradiction with the above.

Theorem 5. Let be a connected balanced bipartite graph with vertices. If , , andfor some integer t, then G is Hamiltonian.

Proof. Since , by Lemma 2, we get . By Lemma 8, G is Hamiltonian or . By the same discussion as the proof of Theorem 4, the conclusion is established.

Theorem 6. Let be a connected balanced bipartite graph of with vertices. If , , andfor some integer t, then G is Hamiltonian.

Proof. Since , by Lemma 3, . By Lemma 8, G is Hamiltonian or . By the same discussion as the proof of Theorem 4, the conclusion is established.

5. Traceable of Nearly Balanced Bipartite Graphs

Lemma 9 (Yu, Fang, and Fan [24]). Let G be a nearly balanced bipartite graph with vertices. If , , andfor some integer t, then G is traceable or .

Theorem 7. Let be a connected nearly balanced bipartite graph with vertices. If , andfor some integer t, then G is traceable.

Proof. Let , where and , thenwhere and . Because , we get . By Lemma 9, we get G is traceable or .
If : because is connected and , we get . It is in contradiction with the above.

Theorem 8. Let be a connected nearly balanced bipartite graph with vertices. If , andfor some integer t, then G is traceable.

Proof. Let , where and , thenwhere and . Because , we get . By Lemma 9, we obtain that G is traceable or . By the same discussion as the proof of Theorem 7, the conclusion is established.

Theorem 9. Let be a connected nearly balanced bipartite graph with vertices. If , andfor some integer t, then G is traceable or .

Proof. Let , where and , thenwhere and . Becausewe get . By Lemma 9, we obtain that G is traceable or . Note that and  > . If , then . The conclusion is established.

6. Traceable of Graphs

Lemma 10 (see [23]). Let G be a graph with vertices, where t is an integer. If and then G is traceable unless or .

Theorem 10. Let be a connected graph with vertices, where t is an integer. If andthen G is traceable.

Proof. Since , by Lemma 4, we get . By Lemma 10, we obtain that G is traceable unless or .
If : note that . Then, if , we have , a contradiction.
If : note that Then, if , we have , a contradiction.

Theorem 11. Let be a connected graph with vertices, where t is an integer. If andthen G is traceable.

Proof. Since  −  +  −  −  by Lemma 5, we get . By Lemma 10, we obtain that G is traceable or or .
If : note that . Then, if , we have  −  +  −  − , a contradiction.
If : note that . Then, if , we have  −  +  −  − , a contradiction.

Theorem 12. Let be a connected graph with vertices, where t is an integer. If andthen G is traceable.

Proof. Since by Lemma 6, we get . By Lemma 10, we obtain that G is traceable unless or .
If : note that Then, if , we have , a contradiction.
If : note that Then, if , we have a contradiction.

7. Hamiltonian of Graphs

Lemma 11. (see [23]). Let G be a graph with vertices, where t is an integer. If andthen G is Hamiltonian or or .

Theorem 13. Let be a connected graph with vertices, where t is an integer. If andthen G is Hamiltonian.

Proof. Since by Lemma 4, we get . By Lemma 11, we obtain that G is Hamiltonian or or .
If : note that . Then, if we have a contradiction.
If : note that . Then, if we have , a contradiction.

Theorem 14. Let and be a connected graph with vertices for some integers t satisfying the following condition:Therefore, G is Hamiltonian.

Proof. Since  −  +  −  − , by Lemma 5, we get . By Lemma 11, we obtain that G is Hamiltonian or or .
If : note that . Then, if we have  −  +  −  − , a contradiction.
If : note that . Then, if we have  −  +  −  − , a contradiction.

Theorem 15. Let be a connected graph with vertices for some integer t. If and , then G is Hamiltonian.

Proof. Since , by Lemma 6, we get . By Lemma 11, we obtain that G is Hamiltonian unless or .
If : note that . Then, if , we have  >  a contradiction.
If : note that . Then, if , we have  >  a contradiction.

8. Hamilton-Connected of Graphs

Lemma 12 (see [20]). Let G be a t-connected graph with n vertices, where . If then G is Hamilton-connected.
By Lemmas 4, 5, 6, and 12 and by direct computations, we get Theorems 16, 17, and 18, respectively.

Theorem 16. Let G be a t-connected graph with n vertices and be a connected graph, where . If then G is Hamilton-connected.

Theorem 17. Let G be a t-connected graph with n vertices and be a connected graph, where . Ifthen G is Hamilton-connected.

Theorem 18. Let G be a t-connected graph with n vertices and be a connected graph, where . Ifthen G is Hamilton-connected.

9. Traceable from Every Vertex of Graphs

Lemma 13 (see [20]). Let G be a t-connected graph with n vertices, where . If then G is traceable from every vertex.
By Lemmas 4, 5, 6, and 13 and by direct computation, we get Theorems 19, 20, and 21, respectively.

Theorem 19. Let G be a t-connected graph with n vertices and be a connected graph, where . Ifthen G is traceable from every vertex.

Theorem 20. Let G be a t-connected graph with n vertices and be a connected graph, where . Ifthen G is traceable from every vertex.

Theorem 21. Let G be a t-connected graph with n vertices and be a connected graph, where . Ifthen G is traceable from every vertex.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was jointly supported by the National Natural Science Foundation of China (grant no. 11871077), the Natural Science Foundation of Anhui Province (grant no. 1808085MA04), and the Natural Science Foundation of Department of Education of Anhui Province (grant no. KJ2017A362).